Chapter 2 USING THE SWITCHABLE QUALITY OF A LIQUID CRYSTAL SOLVENT TO MEDIATE SEGREGATION BETWEEN COIL AND LIQUID CRYSTALLINE HOMOPOLYMERS

Size: px
Start display at page:

Download "Chapter 2 USING THE SWITCHABLE QUALITY OF A LIQUID CRYSTAL SOLVENT TO MEDIATE SEGREGATION BETWEEN COIL AND LIQUID CRYSTALLINE HOMOPOLYMERS"

Transcription

1 22 Chpter 2 USING THE SWITCHABLE QUALITY OF A LIQUID CRYSTAL SOLVENT TO MEDIATE SEGREGATION BETWEEN COIL AND LIQUID CRYSTALLINE HOMOPOLYMERS Chpter Introduction Experimentl Results Discussion Conclusions Tles Figures References...34 Reproduced in prt with permission from Scruggs, Kornfield, nd Ll, Mcromolecules 2006, 39, Copyright 2006 Americn Chemicl Society. 2.1 Introduction The thermodynmics governing phse ehvior in polymer solutions is strikingly different from tht of regulr, smll-molecule solutions. The entropy of mixing tht fvors misciility etween smll-molecule species is sustntilly reduced y the covlent connectivity of monomers in polymer chin. The usully unfvorle enthlpy of mixing two species is, therefore, more difficult to overcome when one of them is polymer. The competition etween entropy nd enthlpy is summrized y the Flory-Huggins eqution [1]

2 23 which expresses the free energy of mixing, F mix, s function of the polymer s volume frction, φ, nd degree of polymeriztion, N, s well s n interction prmeter, χ, tht quntifies the enthlpy of mixing: F mix φ = lnφ + ( 1 φ) ln( 1 φ) + χφ( 1 φ), (2.1) kt N where k is Boltzmnn s constnt nd T is the temperture. The Flory interction prmeter is usully positive nd hs een found empiriclly to depend on temperture ccording to where A nd B re constnts. [1] B χ ( T ) = A +, (2.2) T Smll-molecule nemtic liquid crystls (LCs) re prticulrly interesting s solvents for polymers ecuse they cn undergo first-order trnsition etween ordered (nemtic) nd disordered (isotropic) fluid phses, nd the χ prmeter chnges discontinuously t the nemtic-isotropic trnsition temperture (T NI ). Below T NI the entropic penlty of dissolving rndom-coil polymer in ordered solvent is lumped into the χ prmeter, which grows with the solvent s degree of orienttionl order, ut ove T NI this penlty is lost nd the LC ehves like conventionl solvent. [2] If, however, the polymer lso hs liquid crystlline chrcter, χ my switch from eing positive to negtive when the solvent ecomes isotropic. In either cse, the misciility of polymer with n LC solvent my chnge drsticlly in the smll temperture window round T NI. Rich thermodynmics result from dissolving coil polymer nd side-group liquid crystl polymer (SGLCP) together in n LC solvent. The ternry phse ehvior of such polymer-polymer-solvent system is determined y the entropy of mixing lnced with polymer-solvent interctions (chrcterized y χ AS nd χ BS ) s well s polymer-polymer interctions (chrcterized y χ AB ). [3-7] In nemtic solvent, ll three χ prmeters chnge discontinuously t T NI nd the solution s phse ehvior is ltered ccordingly. In this chpter, the inry phse ehvior of rndom-coil polymer (polystyrene, PS) in n LC solvent (4-pentyl-4 -cynoiphenyl, 5CB) is compred to solutions of n SGLCP in

3 24 5CB in order to correlte polymer properties with chnges in polymer misciility tht tke plce t T NI. These results elucidte the physics ehind the rich phse ehvior of ternry solutions of PS, SGLCP, nd 5CB tht hs consequences for solutions of coil- SGLCP lock copolymers discussed in lter chpters. 2.2 Experimentl Mterils A side-group liquid crystl homopolymer, 490HSiCB4, ws synthesized nd chrcterized ccording to the methods descried in Appendix A (Figure 2.1). A 63 kg/mol 1,2- polyutdiene (PB) prepolymer ws synthesized vi living nionic polymeriztion y Steven Smith of Proctor nd Gmle, Inc. (see Appendix A for chrcteriztion). After ttching the mesogenic side-groups, the polymer s moleculr weight (M n ) is 489 kg/mol nd 97 mol % of the monomers hve SiCB4 mesogens ttched. Of the remining 4 mol %, 1 mol % of the monomers re residul 1,2-utdiene nd 3 mol % re unrective 1,4- utdiene. The polydispersity (PDI = M w / M n ) is 1.48 (Tle 2.1). Monodisperse polystyrene (PS) homopolymer with M n = 44 kg/mol ws purchsed nd used s received from Aldrich. The nemtic LC solvent 4-pentyl-4 -cynoiphenyl (5CB) ws used s received from TCI Americ Methods Ternry mixtures of 490HSiCB4 homopolymer nd PS homopolymer in 5CB were mde y comining the three components in controlled quntities for totl mss of pproximtely 50 mg then dissolving them together in ~100 µl of tetrhydrofurn (THF). Smples were mixed for t lest n hour to ensure complete dissolution of ll three components. A smll drop of the THF solution ws plced in the shllow well of n indented microscope slide nd the THF ws evported wy t elevted temperture (~100 o C). Slides were exmined using Zeiss Universl opticl microscope equipped with Mettler FP82 hot stge nd removle polrizers. Ech smple ws first heted from room temperture t rte of 2 o C/min to oserve the nemtic to isotropic phse trnsition; the temperture t which the colorful, irefringent texture viewed etween

4 25 crossed polrizers (Figure 2.2) disppered ws recorded s the isotropiztion point (T NI ). Next, the polrizers were removed nd the smple ws heted further until droplets chrcteristic of two-phse, isotropic/isotropic coexistence (Figure 2.2) were no longer oserved (Figure 2.2c). Then, the smple ws cooled t rte of 10 o C/min nd the temperture t which droplets reppered ws recorded s nominl upper criticl solution temperture (UCST). This mesurement ws repeted five times, exmining different re on the slide ech time, nd the results verged. To set ound on the sucooling required for oservle drops to form on cooling, severl smples were rised to 5 C ove the nominl UCST. Consistently, single phse formed over time. Therefore, the true UCST lies within 5 C of the nominl one. Since the chnges in UCST with solution composition were very lrge, this uncertinty did not ffect the conclusions of this study. 2.3 Results Binry solutions of 490HSiCB4 with 5CB were oserved to e single phse oth ove nd elow T NI (Figure 2.3). Binry solutions of PS with 5CB exhiit smll iphsic window in the isotropic phse of 7.4 o C or less (Figure 2.3) nd re single-phse ove n upper criticl solution temperture (UCST). This result is consistent with previously estlished phse digrms of PS with 5CB, [8, 9] which found the two to e miscile ove 40 o C, provided the concentrtion of PS is less thn 50 wt %. Ternry mixtures of PS, 490HSiCB4, nd 5CB were oserved to phse seprte elow T NI into coexisting nemtic nd isotropic phses t ll compositions tested (up to 20 wt %). The mesured isotropiztion points in ternry mixtures (i.e., trnsition from N + I to I + I iphse or N + I to I) were found to e within 3.5 o C of the ulk T NI of 5CB (35 o C), indicting tht the nemtic phse contins little or no isotropic diluent nd prtitioning of the polymers into n SGLCP-rich, nemtic phse nd PS-rich, isotropic phse must therefore e nerly complete. Aove T NI wide misciility gps were oserved up to the UCST.

5 26 The UCST of ternry mixtures of PS, 490HSiCB4, nd isotropic 5CB ws found to e highly sensitive to the concentrtions of the two polymers nd n effective soluility limit ws quickly reched with incresing concentrtions (Figure 2.4). For exmple, holding the concentrtion of 490HSiCB4 fixed t 4.5 wt %, the ddition of 0.5 wt % PS opens up 14 o C misciility gp (UCST = 49 o C) which expnds to 97 o C (UCST = 132 o C) when the concentrtion of PS is 6 wt %. When the overll concentrtion of polymer, PS nd 490HSiCB4 comined, exceeds pproximtely 10 wt %, the UCST cnnot e reched efore the smple thermlly degrdes, representing n effective soluility limit. This steep increse in UCST with incresing polymer concentrtion is surprising ecuse ech of the two polymers dissolves in ll proportions tested in isotropic 5CB lone. 2.4 Discussion Binry nd ternry mixtures of 490HSiCB4 nd PS dissolved in 5CB show tht nemtic 5CB is strongly selective solvent for the SGLCP. As previously reported y Hori et l., [9] PS is insolule in nemtic 5CB; mixtures phse seprte into n isotropic, PS-rich phse nd nemtic phse nerly devoid of PS ltogether. On the other hnd, 490HSiCB4 is solule in nemtic 5CB t ll concentrtions tested (up to 20 wt %). The liquid crystlline order of the nemtic solvent imposes lrge entropic penlty to solvtion of rndom coil polymer (e.g. PS), [9-11] ut the chemiclly similr side-groups nd liquid crystlline nture of the SGLCP fcilitte misciility with the nemtic solvent. [2] The oservtion of lrge misciility gps in ternry mixtures of PS, 490HSiCB4, nd 5CB ws surprising ecuse single-phse solutions re esily chieved in inry mixtures of 5CB with either polymer lone. One reson for the polymers poor soluility in ternry solutions is tht unfvorle interctions etween the polymers themselves ( lrge contriution of χ AB ) increse the free energy of mixing. [7] However, this effect is wek in dilute solutions where polymer-polymer interctions re effectively screened y solvent. Smll-ngle neutron scttering experiments presented in Chpter 5 demonstrte tht 5 wt % solutions of similr SGLCPs (310HSiCB4 nd 780HSiCB4) re in the semidilute regime, ut when the concentrtion of 490HSiCB4 is sustntilly lower the polymer chins should

6 27 e dilute nd non-intercting. The overlp concentrtion of PS is clculted [12] to e pproximtely 7 wt % t 35 C in cyclohexne [13] which represents lower ound on the dilute regime since 5CB is not s good solvent for PS s cyclohexne. A portion of the mesured phse digrm is, therefore, well within regime where the free energy contriution of inter-polymer interction (χ AB ) should e smll. Besides polymer-polymer interctions, polymer-solvent interctions lso contriute to the system s thermodynmics, s descried y the Ptterson [6] -Prusnitz [3] tretment of the Scott [4] -Tomp [5] theory for ternry polymer-polymer solutions. Ptterson nd Prusnitz found tht ny differentil preference of the solvent for one polymer over the other (χ AS χ BS ), even if tht preference is smll, cn induce phse seprtion. Furthermore, the effect is excerted when the moleculr weight difference etween the polymers is lrge. The symmetric solvent effect is, therefore, predicted to ply sustntil role in the thermodynmics governing phse seprtion of solutions of PS nd 490HSiCB4 in 5CB: the chemicl structure of the two polymers is sufficiently different to cuse n pprecile difference in polymer-solvent interctions etween the two, nd the moleculr weight rtio of PS to HSiCB4 is pproximtely 1:10. Isotropic 5CB is, therefore, not neutrl solvent (χ AS = χ BS ), ut is etter clssified s slightly selective ( χ AS - χ BS << χ AS χ BS ). Compred to conventionl solvents, 5CB is unique in its ility to undergo discontinuous, thermlly-induced chnge in its misciility with polymers. Solvent qulity is typiclly monotonic function of temperture, [1] ut the first-order nemtic-isotropic phse trnsition results in n rupt chnge so tht incresing temperture less thn 1 C cn cuse n initilly insolule polymer to ecome miscile with 5CB. This phenomenon is termed switchle solvent qulity. 2.5 Conclusions The switchle solvent qulity of n LC solvent results in rich ternry solution thermodynmics. Unfvorle polymer-polymer interctions work together with strong solvent selectivity to drive mcroscopic phse seprtion in the nemtic phse. In the

7 28 isotropic phse, fvorle interctions etween the solvent nd oth polymers re lnced ginst polymer-polymer incomptiility nd the symmetric solvent effect, resulting in phse ehvior tht is strikingly sensitive to chnges in the solution s composition. Modultion etween the two regimes is chieved with temperture chnges of less thn 1 C. The unique phse ehvior of homopolymer solutions in n LC solvent suggests tht the driving force for self-ssemly of coil-sglcp lock copolymer could lso e modulted y the solvent s trnsition etween the nemtic nd isotropic phses. Below T NI the solvent s orienttionl order would drive the lock copolymer to self-ssemle into structure tht segregtes the coil-lock from the LC host. Aove T NI this driving force would e lost, ut self-ssemly could still occur s result of the sme thermodynmic interctions tht drive phse seprtion in the isotropic homopolymer solutions. This topic is explored in detil in Chpter 3.

8 Tles Tle 2.1 Moleculr weight, conversion, nd polydispersity of the side-group liquid crystl homopolymer. Detils of chrcteriztion re presented in Appendix A. Nme M n [kg/mol] Mole Frction 1,2 PB Mole Frction 1,4 PB Mole Frction LC PDI 490HSiCB PDI = Polydispersity Index (M w /M n )

9 Figures 490HSiCB4 PS 5CB Si O Si C N O C N Figure 2.1 Chemicl structures of side-group liquid crystl polymer (490HSiCB4), polystyrene (PS), nd nemtic liquid crystl solvent (5CB). The SGLCP s nme is derived from its moleculr weight (489 kg/mol), the letter H to indicte homopolymer, nd SiCB4 to indicte end-on ttchment of the mesogens. In ddition to monomers hving n ttched mesogen, the polymer lso contins ~ 1 mol % ech of residul 1,2- nd 1,4-utdiene monomers. Its properties re summrized in Tle 2.1 nd the detils of its synthesis nd chrcteriztion re presented in Appendix A.

10 Figure 2.2 Opticl microgrphs typicl of ternry lends of PS, 490HSiCB4, nd 5CB t tempertures () elow T NI, () etween T NI nd the UCST, nd (c) ove UCST. The smple is imged etween crossed polrizers in the nemtic phse (T < T NI ). These prticulr microgrphs re of 2.25 wt % PS nd 4.57 wt % 490HSiCB4 in 5CB t () 34 C, () 70 C, nd (c) 90 C. 31

11 32 () 50 () T ( C) ( C) T ( C) I N T NI HSiCB4 / 5CB φ (wt. frc.) I I+I N+I φ (wt. frc.) UCST T NI PS / 5CB Figure 2.3 Prtil inry phse digrms of () 490HSiCB4 homopolymer in 5CB nd () polystyrene homopolymer with 5CB. Dshed lines re drwn to guide the eye towrd plusile phse oundries. The letters N nd I indicte single nemtic or isotropic phse, respectively. I+I indictes two coexisting isotropic phses nd N+I indictes coexisting nemtic nd isotropic phses.

12 UCST ( o C) c d e f g h i j k l T NI >140 [HSiCB4] (wt. frc.) l d e d [5CB] (wt. frc.) d j j k k [PS] (wt. frc.) h i l l g Cnnot Access Single Phse l 33 Figure 2.4 Prtil ternry phse digrm of PS (44 kg/mol), 490HSiCB4, nd isotropic 5CB determined from opticl microscopy. The shding nd letter of ech point expresses the upper criticl solution temperture (UCST) t which single-phse solution is otined. A UCST equl to T NI mens the solution ecme single-phse immeditely upon trnsitioning to the isotropic phse. A UCST > 140 C mens single phse ws inccessile ecuse the mixture egn to decompose. The dshed line is drwn to guide the eye to the region where single-phse solution cnnot e reched.

13 References [1] Ruinstein, M.; Coly, R. H. Polymer Physics, 1 st ed; Oxford University Press: New York, [2] Brochrd, F. Solutions of flexile polymers in nemtic liquid. C. R. Acd. Sc. Pris 1979, 289, [3] Hsu, C. C.; Prusnitz, J. M. Thermodynmics of Polymer Comptiility in Ternry Systems. Mcromolecules 1974, 7, [4] Scott, R. L. The Thermodynmics of High Polymer Solutions. V. Phse Equiliri in the Ternry System: Polymer 1- Polymer 2- Solvent. J. Chem. Phys. 1949, 17, [5] Tomp, H. Phse Reltionships in Polymer Solutions. Trns. Frdy Soc. 1949, 45, [6] Zemn, L.; Ptterson, D. Effect of the Solvent on Polymer Incomptiility in Solution. Mcromolecules 1972, 5, [7] Olisi, O.; Roeson, L. M.; Shw, M. T. Polymer-Polymer Misciility, Acdemic Press, Inc.: Sn Diego, [8] Hkemi, H. Elstic constnts in dilute poly(styrene)/nemtic liquid crystl solutions - effects of concentrtion nd moleculr weight. Polymer 1999, 40, [9] Hori, H.; Urkw, O.; Adchi, K. Dielectric Relxtion in Phse-Segregted Mixtures of Polystyrene nd Liquid Crystl 5CB. Mcromolecules 2004, 37, [10] Benmoun, F.; Doudi, A.; Roussel, F.; Buisine, J.-M.; Coqueret, X.; Mschke, U. Equilirium Phse Digrm of Polystyrene nd 8CB. J. Polym. Sci., Prt B: Polym. Phys. 1999, 37, [11] Gogius, N.; Mschke, U.; Benmoun, F.; Ewen, B.; Coqueret, X.; Benmoun, M. Phse Digrms of Poly(dimethylsiloxne) nd 5CB Blends. J. Polym. Sci., Prt B: Polym. Phys. 2001, 39, [12] Nod, I.; Kto, N.; Kitno, T.; Ngsw, M. Thermodynmics of Modertely Concentrted Solutions of Liner Polymers. Mcromolecules 1981, 14, [13] Brndup, J.; Immergut, E. H.; Grulke, E. A., eds. Polymer Hndook. 4 th ed. 1999, Wiley: New York.

WILL A POLYMER DISSOLVE IN A GIVEN SOLVENT? WILL A PARTICULAR POLYMER MIX WITH ANOTHER POLYMER?

WILL A POLYMER DISSOLVE IN A GIVEN SOLVENT? WILL A PARTICULAR POLYMER MIX WITH ANOTHER POLYMER? MIXING WILL POLYMER DISSOLVE IN GIVEN SOLVENT? WILL PRTICULR POLYMER MIX WITH NOTHER POLYMER? POLYMER SOLUTIONS ND LENDS MISCILE IMMISCILE lends Single phase Phase separated Solvent (or polymer) molecules

More information

Module 2: Rate Law & Stoichiomtery (Chapter 3, Fogler)

Module 2: Rate Law & Stoichiomtery (Chapter 3, Fogler) CHE 309: Chemicl Rection Engineering Lecture-8 Module 2: Rte Lw & Stoichiomtery (Chpter 3, Fogler) Topics to be covered in tody s lecture Thermodynmics nd Kinetics Rection rtes for reversible rections

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

2.4 Linear Inequalities and Interval Notation

2.4 Linear Inequalities and Interval Notation .4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

More information

Homework Assignment 6 Solution Set

Homework Assignment 6 Solution Set Homework Assignment 6 Solution Set PHYCS 440 Mrch, 004 Prolem (Griffiths 4.6 One wy to find the energy is to find the E nd D fields everywhere nd then integrte the energy density for those fields. We know

More information

Rates of chemical reactions

Rates of chemical reactions Rtes of chemicl rections Mesuring rtes of chemicl rections Experimentl mesuring progress of the rection Monitoring pressure in the rection involving gses 2 NO( g) 4 NO ( g) + O ( g) 2 5 2 2 n(1 α) 2αn

More information

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length

BOND ORDER (BO): Single bond Þ BO = 1; Double bond Þ BO = 2; Triple bond Þ BO = 3 Bond Order Þ bond strength and bond length EMISTRY 104 elp Sheet #1 hem 103 Review (Text: h 6, h 7) Do topics pproprite for your lecture Prepred y Dr. Tony Jco http://www.chem.wisc.edu/res/clc (Resource pge) Nuggets: Electronegtivity (6.7), Bond

More information

Industrial Electrical Engineering and Automation

Industrial Electrical Engineering and Automation CODEN:LUTEDX/(TEIE-719)/1-7/(7) Industril Electricl Engineering nd Automtion Estimtion of the Zero Sequence oltge on the D- side of Dy Trnsformer y Using One oltge Trnsformer on the D-side Frncesco Sull

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

1B40 Practical Skills

1B40 Practical Skills B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need

More information

Psychrometric Applications

Psychrometric Applications Psychrometric Applictions The reminder of this presenttion centers on systems involving moist ir. A condensed wter phse my lso be present in such systems. The term moist irrefers to mixture of dry ir nd

More information

ANALYSIS OF FAST REACTORS SYSTEMS

ANALYSIS OF FAST REACTORS SYSTEMS ANALYSIS OF FAST REACTORS SYSTEMS M. Rghe 4/7/006 INTRODUCTION Fst rectors differ from therml rectors in severl spects nd require specil tretment. The prsitic cpture cross sections in the fuel, coolnt

More information

Convert the NFA into DFA

Convert the NFA into DFA Convert the NF into F For ech NF we cn find F ccepting the sme lnguge. The numer of sttes of the F could e exponentil in the numer of sttes of the NF, ut in prctice this worst cse occurs rrely. lgorithm:

More information

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5.

Reading from Young & Freedman: For this topic, read the introduction to chapter 24 and sections 24.1 to 24.5. PHY1 Electricity Topic 5 (Lectures 7 & 8) pcitors nd Dielectrics In this topic, we will cover: 1) pcitors nd pcitnce ) omintions of pcitors Series nd Prllel 3) The energy stored in cpcitor 4) Dielectrics

More information

CHAPTER 20: Second Law of Thermodynamics

CHAPTER 20: Second Law of Thermodynamics CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het

More information

Acid-Base Equilibria

Acid-Base Equilibria Tdeusz Górecki Ionic Equiliri Acid-Bse Equiliri Brønsted-Lory: n cid is proton, se is. Acid Bse ( 3 PO 4, O), ( N 4 ) nd ( PO - 4 ) cn ll ehve s cids. Exmple: 4 N N3 Sustnces hich cn ehve oth s cids nd

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Which of the following describes the net ionic reaction for the hydrolysis. Which of the following salts will produce a solution with the highest ph?

Which of the following describes the net ionic reaction for the hydrolysis. Which of the following salts will produce a solution with the highest ph? 95. Which of the following descries the net ionic rection for the hydrolysis of NH4Cl( s)? A. NH4 ( q) Cl & ( q) NH4Cl( s) B. NH Cl & 4 ( s) NH4 ( q) Cl ( q) C. Cl ( q) H O & 2 ( l) HCl( q) OH ( q) D.

More information

The Properties of Stars

The Properties of Stars 10/11/010 The Properties of Strs sses Using Newton s Lw of Grvity to Determine the ss of Celestil ody ny two prticles in the universe ttrct ech other with force tht is directly proportionl to the product

More information

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

Rel Gses 1. Gses (N, CO ) which don t obey gs lws or gs eqution P=RT t ll pressure nd tempertures re clled rel gses.. Rel gses obey gs lws t extremely low pressure nd high temperture. Rel gses devited

More information

CBE 291b - Computation And Optimization For Engineers

CBE 291b - Computation And Optimization For Engineers The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn

More information

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

More information

Physics 1402: Lecture 7 Today s Agenda

Physics 1402: Lecture 7 Today s Agenda 1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry

LECTURE 14. Dr. Teresa D. Golden University of North Texas Department of Chemistry LECTURE 14 Dr. Teres D. Golden University of North Texs Deprtment of Chemistry Quntittive Methods A. Quntittive Phse Anlysis Qulittive D phses by comprison with stndrd ptterns. Estimte of proportions of

More information

Chapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis

Chapter 4: Techniques of Circuit Analysis. Chapter 4: Techniques of Circuit Analysis Chpter 4: Techniques of Circuit Anlysis Terminology Node-Voltge Method Introduction Dependent Sources Specil Cses Mesh-Current Method Introduction Dependent Sources Specil Cses Comprison of Methods Source

More information

Silicon Nanowire Based Single-Molecule SERS Sensor

Silicon Nanowire Based Single-Molecule SERS Sensor Supporting informtion Silicon Nnowire Bsed Single-Molecule SERS Sensor Hui Wng, Xuemei Hn, Xuemei Ou, Chun-Sing Lee, Xiohong Zhng* nd Shuit-Tong Lee S1, A series of Si nnowires coted with compct ggregtes

More information

Data Provided: A formula sheet and table of physical constants is attached to this paper. THE PHYSICS OF SOFT CONDENSED MATTER

Data Provided: A formula sheet and table of physical constants is attached to this paper. THE PHYSICS OF SOFT CONDENSED MATTER Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: Grph pper (liner) DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn Semester (2015) THE PHYSICS OF SOFT CONDENSED

More information

Section 6: Area, Volume, and Average Value

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

More information

Exercises with (Some) Solutions

Exercises with (Some) Solutions Exercises with (Some) Solutions Techer: Luc Tesei Mster of Science in Computer Science - University of Cmerino Contents 1 Strong Bisimultion nd HML 2 2 Wek Bisimultion 31 3 Complete Lttices nd Fix Points

More information

The Thermodynamics of Aqueous Electrolyte Solutions

The Thermodynamics of Aqueous Electrolyte Solutions 18 The Thermodynmics of Aqueous Electrolyte Solutions As discussed in Chpter 10, when slt is dissolved in wter or in other pproprite solvent, the molecules dissocite into ions. In queous solutions, strong

More information

Derivations for maximum likelihood estimation of particle size distribution using in situ video imaging

Derivations for maximum likelihood estimation of particle size distribution using in situ video imaging 2 TWMCC Texs-Wisconsin Modeling nd Control Consortium 1 Technicl report numer 27-1 Derivtions for mximum likelihood estimtion of prticle size distriution using in situ video imging Pul A. Lrsen nd Jmes

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

Chapter 6 Techniques of Integration

Chapter 6 Techniques of Integration MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln

More information

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations

Fig. 1. Open-Loop and Closed-Loop Systems with Plant Variations ME 3600 Control ystems Chrcteristics of Open-Loop nd Closed-Loop ystems Importnt Control ystem Chrcteristics o ensitivity of system response to prmetric vritions cn be reduced o rnsient nd stedy-stte responses

More information

M344 - ADVANCED ENGINEERING MATHEMATICS

M344 - ADVANCED ENGINEERING MATHEMATICS M3 - ADVANCED ENGINEERING MATHEMATICS Lecture 18: Lplce s Eqution, Anltic nd Numericl Solution Our emple of n elliptic prtil differentil eqution is Lplce s eqution, lso clled the Diffusion Eqution. If

More information

Joule-Thomson effect TEP

Joule-Thomson effect TEP Joule-homson effect EP elted oics el gs; intrinsic energy; Gy-Lussc theory; throttling; n der Wls eqution; n der Wls force; inverse Joule- homson effect; inversion temerture. Princile A strem of gs is

More information

Temperature influence compensation in microbolometer detector for image quality enhancement

Temperature influence compensation in microbolometer detector for image quality enhancement .26/qirt.26.68 Temperture influence compenstion in microolometer detector for imge qulity enhncement More info out this rticle: http://www.ndt.net/?id=2647 Astrct y M. Krupiński*, T. Sosnowski*, H. Mdur*

More information

2. VECTORS AND MATRICES IN 3 DIMENSIONS

2. VECTORS AND MATRICES IN 3 DIMENSIONS 2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

More information

Lecture 08: Feb. 08, 2019

Lecture 08: Feb. 08, 2019 4CS4-6:Theory of Computtion(Closure on Reg. Lngs., regex to NDFA, DFA to regex) Prof. K.R. Chowdhry Lecture 08: Fe. 08, 2019 : Professor of CS Disclimer: These notes hve not een sujected to the usul scrutiny

More information

Section 4: Integration ECO4112F 2011

Section 4: Integration ECO4112F 2011 Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

More information

CHEMICAL KINETICS

CHEMICAL KINETICS CHEMICAL KINETICS Long Answer Questions: 1. Explin the following terms with suitble exmples ) Averge rte of Rection b) Slow nd Fst Rections c) Order of Rection d) Moleculrity of Rection e) Activtion Energy

More information

Dynamic equilibrium occurs when the forward and reverse reactions occur at the same rate.

Dynamic equilibrium occurs when the forward and reverse reactions occur at the same rate. MODULE 2 WORKSHEET8 EQUILIBRIUM Syllus reference 9.3.2 1 Clssify ech of the following sttements s true or flse. For the flse sttements rewrite them so they re true. For chemicl equilirium to e estlished

More information

QUADRATURE is an old-fashioned word that refers to

QUADRATURE is an old-fashioned word that refers to World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd

More information

The development of nanoscale morphology in polymer:fullerene. photovoltaic blends during solvent casting

The development of nanoscale morphology in polymer:fullerene. photovoltaic blends during solvent casting Supplementry informtion Supplementry Mteril (ES) for Soft Mtter The development of nnoscle morphology in polymer:fullerene photovoltic lends during solvent csting To Wng, * Aln D. F. Dunr, Pul A. Stniec,

More information

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry

More information

Satellite Retrieval Data Assimilation

Satellite Retrieval Data Assimilation tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl Artist depiction of NAA terr

More information

Hints for Exercise 1 on: Current and Resistance

Hints for Exercise 1 on: Current and Resistance Hints for Exercise 1 on: Current nd Resistnce Review the concepts of: electric current, conventionl current flow direction, current density, crrier drift velocity, crrier numer density, Ohm s lw, electric

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION Synthesis of metl oxide with roomtemperture photoreversile phse trnsition Shin-ichi Ohkoshi 1 *, Yoshihide Tsunouchi, 1 Tomoyuki Mtsud, 1 Kzuhito Hshimoto, 2 Asuk Nmi, 1 Fumiyoshi

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Worked out examples Finite Automata

Worked out examples Finite Automata Worked out exmples Finite Automt Exmple Design Finite Stte Automton which reds inry string nd ccepts only those tht end with. Since we re in the topic of Non Deterministic Finite Automt (NFA), we will

More information

CHAPTER 08: MONOPROTIC ACID-BASE EQUILIBRIA

CHAPTER 08: MONOPROTIC ACID-BASE EQUILIBRIA Hrris: Quntittive Chemicl Anlysis, Eight Edition CHAPTER 08: MONOPROTIC ACIDBASE EQUILIBRIA CHAPTER 08: Opener A CHAPTER 08: Opener B CHAPTER 08: Opener C CHAPTER 08: Opener D CHAPTER 08: Opener E Chpter

More information

6. Photoionization of acridine through singlet and triplet channels

6. Photoionization of acridine through singlet and triplet channels Chpter 6: Photoioniztion of cridine through singlet nd triplet chnnels 59 6. Photoioniztion of cridine through singlet nd triplet chnnels Photoioinztion of cridine (Ac) in queous micelles hs not yet een

More information

Lecture Solution of a System of Linear Equation

Lecture Solution of a System of Linear Equation ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

More information

4 Results and Discussions

4 Results and Discussions 4 Results nd Discussions 4. Generl Considertion In this chpter, results will be presented nd discussed. Further, if possible, they will be compred with men-field results. As mentioned in previous chpters,

More information

MODIFICATION OF VANT HOFF PARAMETER FOR CONCENTRATE SALT SOLUTION

MODIFICATION OF VANT HOFF PARAMETER FOR CONCENTRATE SALT SOLUTION VOL. 13, NO. 9, MAY 018 ISSN 18196608 ARN Journl of Engineering nd Applied Sciences 006018 Asin Reserch ulishing Network (ARN). All rights reserved. www.rpnjournls.com MODIFICATION OF VANT HOFF ARAMETER

More information

The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases.

The Predom module. Predom calculates and plots isothermal 1-, 2- and 3-metal predominance area diagrams. Predom accesses only compound databases. Section 1 Section 2 The module clcultes nd plots isotherml 1-, 2- nd 3-metl predominnce re digrms. ccesses only compound dtbses. Tble of Contents Tble of Contents Opening the module Section 3 Stoichiometric

More information

temperature is known as ionic product of water. It is designated as K w. Value of K w

temperature is known as ionic product of water. It is designated as K w. Value of K w Ionic product of ter The product of concentrtions of H nd OH ions in ter t prticulr temperture is knon s ionic product of ter. It is designted s K. H O H 1 OH ; H 57.3 kjm The vlue of [H ][OH ] K ; K[HO]

More information

ANSWERS TO REVIEW PROBLEM SET CH F

ANSWERS TO REVIEW PROBLEM SET CH F ANSWERS T REVIEW PRBLEM SET C41-018F [1] () The chirl isomer is stilized y intrmoleculr hydrogen onding etween the two guche groups. Such n rrngement lso keeps the ulky tert-utyl groups frther prt s they

More information

Chapter E - Problems

Chapter E - Problems Chpter E - Prolems Blinn College - Physics 2426 - Terry Honn Prolem E.1 A wire with dimeter d feeds current to cpcitor. The chrge on the cpcitor vries with time s QHtL = Q 0 sin w t. Wht re the current

More information

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system

AQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex

More information

( ) as a fraction. Determine location of the highest

( ) as a fraction. Determine location of the highest AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if

More information

Lecture 6: Diffusion and Reaction kinetics

Lecture 6: Diffusion and Reaction kinetics Lecture 6: Diffusion nd Rection kinetics 1-1-1 Lecture pln: diffusion thermodynmic forces evolution of concentrtion distribution rection rtes nd methods to determine them rection mechnism in terms of the

More information

Supplementary Information

Supplementary Information Supplementry Informtion Spontneously mplified homochirl orgnic-inorgnic nno-helix complex vi self-prolifertion Hlei Zhi, Yn Qun, Li Li, Xing-Yng Liu, Xurong Xu c nd Ruikng Tng*,c Centre for Biomterils

More information

Topics Covered AP Calculus AB

Topics Covered AP Calculus AB Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

More information

Designing Information Devices and Systems I Anant Sahai, Ali Niknejad. This homework is due October 19, 2015, at Noon.

Designing Information Devices and Systems I Anant Sahai, Ali Niknejad. This homework is due October 19, 2015, at Noon. EECS 16A Designing Informtion Devices nd Systems I Fll 2015 Annt Shi, Ali Niknejd Homework 7 This homework is due Octoer 19, 2015, t Noon. 1. Circuits with cpcitors nd resistors () Find the voltges cross

More information

Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

More information

AB Calculus Review Sheet

AB Calculus Review Sheet AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

4.6 Numerical Integration

4.6 Numerical Integration .6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Minimal DFA. minimal DFA for L starting from any other

Minimal DFA. minimal DFA for L starting from any other Miniml DFA Among the mny DFAs ccepting the sme regulr lnguge L, there is exctly one (up to renming of sttes) which hs the smllest possile numer of sttes. Moreover, it is possile to otin tht miniml DFA

More information

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).

( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x). AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f

More information

Always end in a group Forms branches, not crosslinks

Always end in a group Forms branches, not crosslinks 10.569 Synthesis of Polymers Prof. Pul Hmmond Lecture 8: Network Formtion, Sttisticl Approch, Pw Bsed, A Word on MWD for Nonliner Polymeriztion Network formtion Consider first cse: + [ 3 ] monomers so

More information

NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL

NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL NUMERICAL SIMULATION OF FRONTAL MIXED CLOUD SYSTEMS AND CLOUD MICROSTRUCTURE EFFECT ON SATELLITE SIGNAL V. Bkhnov, O. Kryvook, B. Dormn Ukrinin Hydrometeorologicl Reserch Institute, Avenue of Science 37,

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

dy ky, dt where proportionality constant k may be positive or negative

dy ky, dt where proportionality constant k may be positive or negative Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

More information

MIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model:

MIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model: 1 2 MIXED MODELS (Sections 17.7 17.8) Exmple: Suppose tht in the fiber breking strength exmple, the four mchines used were the only ones of interest, but the interest ws over wide rnge of opertors, nd

More information

Acceptance Sampling by Attributes

Acceptance Sampling by Attributes Introduction Acceptnce Smpling by Attributes Acceptnce smpling is concerned with inspection nd decision mking regrding products. Three spects of smpling re importnt: o Involves rndom smpling of n entire

More information

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present?

Strategy: Use the Gibbs phase rule (Equation 5.3). How many components are present? University Chemistry Quiz 4 2014/12/11 1. (5%) Wht is the dimensionlity of the three-phse coexistence region in mixture of Al, Ni, nd Cu? Wht type of geometricl region dose this define? Strtegy: Use the

More information

Chapter 3. Chapter Introduction Experimental Results Discussion Conclusions Tables...

Chapter 3. Chapter Introduction Experimental Results Discussion Conclusions Tables... 35 Chapter 3 SELF-ASSEMBLY OF COIL-LIQUID CRYSTALLINE DIBLOCK COPOLYMERS IN A SOLVENT OF SWITCHABLE QUALITY: STRUCTURE AND INTERACTIONS MEASURED BY SANS AND RHEOMETRY Chapter 3... 35 3.1 Introduction...36

More information

Designing Information Devices and Systems I Discussion 8B

Designing Information Devices and Systems I Discussion 8B Lst Updted: 2018-10-17 19:40 1 EECS 16A Fll 2018 Designing Informtion Devices nd Systems I Discussion 8B 1. Why Bother With Thévenin Anywy? () Find Thévenin eqiuvlent for the circuit shown elow. 2kΩ 5V

More information

Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report

Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block

More information

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17 EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

More information

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2 1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

More information

Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning

Solution for Assignment 1 : Intro to Probability and Statistics, PAC learning Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (

More information

The steps of the hypothesis test

The steps of the hypothesis test ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

Factors affecting the phonation threshold pressure and frequency

Factors affecting the phonation threshold pressure and frequency 3SC Fctors ffecting the phontion threshold pressure nd frequency Zhoyn Zhng School of Medicine, University of Cliforni Los Angeles, CA, USA My, 9 57 th ASA Meeting, Portlnd, Oregon Acknowledgment: Reserch

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE

DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING MOMENT INTERACTION AT MICROSCALE Determintion RevAdvMterSci of mechnicl 0(009) -7 properties of nnostructures with complex crystl lttice using DETERMINATION OF MECHANICAL PROPERTIES OF NANOSTRUCTURES WITH COMPLEX CRYSTAL LATTICE USING

More information

Network Analysis and Synthesis. Chapter 5 Two port networks

Network Analysis and Synthesis. Chapter 5 Two port networks Network Anlsis nd Snthesis hpter 5 Two port networks . ntroduction A one port network is completel specified when the voltge current reltionship t the terminls of the port is given. A generl two port on

More information